We concentrate in this work on the two-step phase transition scenario as described in section REF .
Other types of multi-step phase transitions can be found in recent works [1]}, [2]}, [3]}.
The requirement of a large mixing angle to trigger a sufficiently strong EWPT is crucial in the two-step phase transition.
The one-step phase transition, \((0,~0)\rightarrow (v(T),~w(T))\) , can be stronger, and for models with a mixing angle \(\lesssim 0.2\) , the phase transition may be detectable in planned space-based interferometers [2]}.
We also note that in our analysis, we have restricted the heavy scalar mass \(m_{\mathcal {S}}\)  to be in the range of \(150-2000\)  GeV (\(m_{\mathcal {S}}>m_{\mathcal {H}}\) ),
the strong EWPT with a lower scalar mass, \(m_{\mathcal {S}}<m_{\mathcal {H}}\) , can be found in Refs. [3]}, [6]}.
Future precision Higgs measurements at collider experiments, such as the high-luminosity LHC , the International Linear Collider, and the Circular Electron-Positron Collider, could further probe the mixing angle regions that trigger a strong first-order EWPT
in our scenario [7]}. A more systematic and general analysis of the EWPT and the related collider phenomenology has been given in a recent work [8]}. Our scenario for EWPT and its phenomena at the colliders could be a particular illustration of those more general considerations.
